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Spin Interactions in Binary Black Hole Initial Data

Learn about interactions in binary black hole systems through advanced numerical methods by Scott Hawley. Explore the intricate details of solving constraint equations, boundary conditions, and multigrid techniques in this groundbreaking research.

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Spin Interactions in Binary Black Hole Initial Data

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  1. Spin Interactions in Binary Black Hole Initial Data Scott Hawley*, Richard Matzner, Michael Vitalo Center for Relativity University of Texas at Austin

  2. Scott Hawley, Numerical Relativity 2005, Nov 4 Physical System • York-Lichnerowitz conformal-traceless method. • Background metric and extrinsic curvature • Related to the physical quantities by...

  3. Scott Hawley, Numerical Relativity 2005, Nov 4 Physical System, cont’d • We solve the constraint equations.... ...solve for f, wi

  4. Scott Hawley, Numerical Relativity 2005, Nov 4 Boundary Conditions • Dirichlet at inner boundary (edge of excision mask): • Robin at outer boundary:

  5. Scott Hawley, Numerical Relativity 2005, Nov 4 Multigrid Overview 2:1 grid spacing 2:1 grid spacing • Traditional relaxation methods of solving elliptic problems are extremely inefficient at eliminating long-wavelength components of error, but are very efficient at eliminating short-wavelength components. • They can thus be regarded as smoothing operators (for the error) • Multigrid (Brandt 1978), an O(N) method, basically* works by applying smoothing operators successively at different resolutions to “smooth away” different wavelength-components of the error. The “V-Cycle” Smooth Smooth Restrict Smooth Smooth Prolong SOLVE Prolong (special interp) Restrict *also modify RHS, & prolong doesn’t overwrite

  6. Scott Hawley, Numerical Relativity 2005, Nov 4 Choice of Background: “Binary Kerr-Schild” Matzner, Huq, Shoemaker, 1998 • Kerr-Schild metric is • This suggests a superposition of N BHs via...

  7. Scott Hawley, Numerical Relativity 2005, Nov 4 How ‘Close’ is Background?

  8. Scott Hawley, Numerical Relativity 2005, Nov 4 Binding Energy • Contribution due to spin is... (Wald 1972) We’ll see...

  9. Scott Hawley, Numerical Relativity 2005, Nov 4 Code Details • Elliptic solver. Parallel multigrid code “TEXMEX” • Handles excision (Hawley & Matzner 2004) • In-house, standalone code, designed as a “black box” elliptic solver: input background/guess, output quantities which satisfy constraints • Vertex-centered • Similar to other multigrid solvers (e.g., Brown, Pretorius, Brügmann) • but with unique prolongation/restriction scheme near inner boundary • Smoothing via Newton-Gauss-Seidel, using “rainbow” (like red-black) ordering • Inner boundary values supplied via ...see next slide • SOR on coarsest-grid solves (switches to NGS near target tol.) • Includes separate, 4th- or 2nd-order independent residual evaluator • Use Thornburg’s AH finder for post-process analysis • Have Carpet interface, but it’s out of date • Still testing FMR, Fisheye...

  10. Scott Hawley, Numerical Relativity 2005, Nov 4 Handling Inner Boundaries (i.e. Excision) Definition: Inner boundary points are those points which are immediately interior to a circle of radius rex. Here we show a fine grid and a coincident coarse grid: Restriction scheme: Use weighted restriction everywhere except when doing so would make use of fine-grid inner boundary data. Instead of using IB data, “just copy”. Filling coarse-grid inner boundary values: copy where possible, otherwise via weighted multi-directional extrapolation (inward) from nearby fine grid points

  11. Scott Hawley, Numerical Relativity 2005, Nov 4 Test: Schwarzschild Background

  12. Scott Hawley, Numerical Relativity 2005, Nov 4 Test: BBH Convergence

  13. Scott Hawley, Numerical Relativity 2005, Nov 4 Results... a1 a2 q d • Study binding energy as function of BBH spins • Typical runs: • Spins a1=a2=0.5, d=10M. Hold q2 const, vary q1 • 5133 grids (4 multigrid levels: 653, 1293, 2563 & 5133) • 32 Processors. Bkgrnd takes few mins, solver 3 hrs. • Domain: -15M to 15M, i.e. Dx=M/17. • Excision radius 0.9M • MADM evaluated at 12M

  14. Scott Hawley, Numerical Relativity 2005, Nov 4 Conformal Factor f • a1=a2=0.5 • d=10M • q1 = q2 = 0

  15. Scott Hawley, Numerical Relativity 2005, Nov 4 Binding Energy vs. Spin Orientation Error in bkgnd script? Green

  16. Scott Hawley, Numerical Relativity 2005, Nov 4 Future Work Mesh refinement. 2-level works well: 3-level is OK... • Fix angle defn. Verifyl-3 dependence by watching how amplitude of cosine scales with d. Find where Wald’s eq. breaks down. • Fisheye trans. is coded, untested. • Carpet interface needs rewriting • Parallel performance...? (Mike) • Break grids in two... (Meghann) • Use in constrained evolutions • Force Mike to find ISCOs?

  17. Scott Hawley, Numerical Relativity 2005, Nov 4 Conclusions • Elliptic solver solves constraints to 2nd order • Binding energy measurements agree with “S dot S” part of Wald’s formula • Our use of of rotation angle phi is flawed (error in background-definition script?) • Solver shows promise for several future “physics” applications

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